Closed-Form Expressions of Multilayered Planar Green's Functions That Account for the Continuous Spectrum in the Far Field

The rational function fitting method has been found useful in the derivation of closed-form expressions of spatial-domain Green's functions for multilayered media. However, former implementations of the rational function fitting method lead to Green's functions expressions that are not accurate in the far field when this far field is dominated by the continuous spectrum instead of being dominated by surface waves (as it happens, for instance, in the case of lossy multilayered media). In this paper, the authors introduce a novel implementation of the rational function fitting method, which leads to Green's functions expressions that are accurate in the far field when this is dominated either by the continuous spectrum or by surface waves. In the new approach, the far-field contribution of the continuous spectrum to the Green's functions is numerically fitted in terms of functions with closed-form Hankel transforms, and this far-field contribution is explicitly added to the total least squares approximations of the Green's functions. The numerical results obtained for the Green's functions with the new approach have been compared with numerical results obtained via direct numerical integration of Sommerfeld integrals, and excellent agreement has been found despite the contribution - continuous spectrum or surface waves - dominating the far field.